- Suppose that \( X \) is a connected and compact Riemann surface and let \( \displaystyle f : X \longrightarrow \mathbb{C} \) be a holomorphic function. Show that \( \displaystyle f \) is constant.

- Let \( \displaystyle f : \mathbb{C} \longrightarrow \mathbb{C} \) be a holomorphic bounded function. Show that there is a unique (holomorphic) extension \( \displaystyle \hat{f} : \hat{ \mathbb{C} } \longrightarrow \hat{ \mathbb{C} } \) and then conclude Liouville's theorem (regarding \( \mathbb{C} \)).

Note that \( \displaystyle \hat{ \mathbb{C} } = \mathbb{C} \cup \{ \infty \} \).